We report soft X-ray spectra (4–18 nm) produced by the interaction of 25–100-fs laser pulses at an intensity of up to 7 × 1016 W/cm2 with a cryogenically cooled Ar gas jet. New spectral lines from Ar8+, Ar9+, and Ar10+ charge states appeared with decreasing preexpansion gas temperature. A nonlinear increase of X-ray line emission from Ar7+, Ar8+, and Ar9+ was observed with cooling, which saturated below a certain temperature. The drastic change in the spectrum is attributed to efficient collisional heating and collisional ionization of growing, small- to medium-sized (102–103 atoms) Ar clusters from the cooled jet. When the laser pulse was extended from 25 to 100 fs we observed considerably stronger emission on lines from high charge states, such as Ar8+, Ar9+, and Ar10+, which suggests that the resonance absorption condition could be reached for the 100-fs pulse.